180 problems

Hilbert–Smith conjecture. Every faithful action of a locally compact group on a connected manifold has a Lie group as its acting group.

Newelski's conjecture. In good circumstances, for example when $T$ is an NIP theory, the canonical epimorphism

Lindenstrauss–Tsukamoto conjecture. If

Let $(X,T)$ be a topological dynamical system, meaning that $(X,d)$ is a compact metric space and $T\colon X\to X$ is continuous. Assume that $(X,T)$ has zero topological entropy,…

For a polynomial interval map with real critical points, let an isentropic level set be a set of maps having the same topological entropy. Milnor's monotonicity of entropy conjectu…

Let $(X,T)$ be a topological dynamical system, and let $\mathcal{U}$ be a finite open cover of $X$. Define $p^*_{X,\mathcal{U}}(n)$ as the maximal, over all $n$-element subsets…

Definable topological amenability conjecture. If $G$ is definably topologically amenable, then

A t.d.s. $(X,T)$ is a topological dynamical system, and positive entropy means that its topological entropy is positive. It has DC2 chaos when there is an uncountable set…

Let $h$ be a homeomorphism with module $d$, and let $W_{0 \rightarrow \infty}$ and $W_{\infty \rightarrow 0}$ denote the corresponding dynamical sets. A heteroclinic branch is an…

Let $(X,\mathcal B)$ be a standard Borel space, let $Aut(X,\mathcal B)$ denote its automorphism group equipped with the topology $p$, and let $\mathcal Sm$ be the class of smooth a…

Let $L_p$ be the usual $L_p$ space, with $1\leq p<\infty$, and equip its isometry group $\operatorname{Iso}(L_p)$ with the strong operator topology. Extreme amenability conjecture.…

Let $P$ be the pseudoarc, the unique hereditarily indecomposable chainable continuum, and let $G=H(P)$ be its homeomorphism group. Let $M_G$ denote the universal minimal compact…

Let $m\in\mathbb N$, and let a minimal $\mathbb Z$-system mean a minimal dynamical system with a continuous $\mathbb Z$-action. Its maximal equicontinuous factor is the maximal fac…

Let $G=\operatorname{Homeo}(P)$ be the homeomorphism group of the pseudo-arc $P$, acting on $P$ by evaluation. A universal minimal flow of $G$ is a minimal compact Hausdorff $G$-fl…

The conjecture.

Let $G$ be a group. A set $R\subset G$ is a set of measurable recurrence if, for every measure-preserving $G$-system $(X,\mathcal{X},\mu,(T_g))$ and every $A\in\mathcal{X}$ with…

Almost-finiteness conjecture. Every free action of an amenable group on the Cantor set is almost finite.

Let $A$ be a sequence of integers. We call $A$ an $R$-sequence for totally minimal systems if, for every totally minimal system $(X,T)$, there exists $x\in X$ such that the orbit o…

Leibman's conjecture. The orbit closure

Glasner–Huang–Shao–Weiss–Ye conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Projective compactification no-retraction conjecture. Every projective $\Gamma$-compactification with non-trivial boundary $X$ is non-Dirac. In particular, there does not exist a r…

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $(X,S)$ be a bounded speedup of $(X,T)$ with orbit number $c$. For the Kakutani–Rokhlin partition…

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $c$ be a positive integer. A minimal speedup with orbit number $c$ means a minimal bounded speedup $(X,S)$ of…

Let $(X,T)$ be a Toeplitz flow and let $(X,S)$ be a minimal bounded speedup, with $S(x)=T^{p(x)}(x)$. Suppose a Kakutani–Rokhlin partition has a common cyclic orbit permutation and…

Let $(X,T)$ be a Toeplitz flow, meaning a symbolic, minimal, almost one-to-one extension of an odometer, and let $(X,S)$ be its minimal bounded speedup, where $S(x)=T^{p(x)}(x)$ fo…